Math 360, Fall 2019, Assignment 8

Mathematical proofs, like diamonds, are hard as well as clear, and will be touched with nothing but strict reasoning.

- John Locke, Second Reply to the Bishop of Worcester

Read:

1. Section 6.

Carefully define the following terms, then give one example and one non-example of each:

1. $\mathrm{gcd}(a,b)$.
2. $\mathrm{lcm}(a,b)$.

Carefully state the following theorems (you need not prove them):

1. Classification of subgroups of cyclic groups ("Every subgroup of a cyclic group is...").
2. Containment criterion for subgroups of $\mathbb{Z}$.
3. Equality criterion for subgroups of $\mathbb{Z}$.
4. Classification of subgroups of $\mathbb{Z}$ ("Every subgroup of $\mathbb{Z}$ has a unique...").
5. Theorem concerning the properties of $\mathrm{gcd}(a,b)$.
6. Theorem concerning the properties of $\mathrm{lcm}(a,b)$.
7. Theorem relating $\mathrm{gcd}(a,b)$ to $\mathrm{lcm}(a,b)$.
8. Containment criterion for subgroups of $\mathbb{Z}_n$.
9. Theorem relating $\left\langle[a]\right\rangle$ to $\left\langle[\mathrm{gcd}(a,n)]\right\rangle$ (in $\mathbb{Z}_n$).
10. Classification of subgroups of $\mathbb{Z}_n$ ("Every subgroup of $\mathbb{Z}_n$ is generated by a unique...").

Solve the following problems:

1. Section 6, problems 5, 7, 22, 23, and 24.
2. In some of the problems above, you drew subgroup diagrams for $\mathbb{Z}_n$ for various values of $n$. Now try to draw the subgroup diagram for $\mathbb{Z}$.

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